Bounded Derivations on Uniform Roe Algebras

On the uniform Roe algebra as a Banach algebra and embeddings of $\ell_p$ uniform Roe algebras

We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe $\mathrm{C}^*$-algebras which depend only on their Banach algebra structure.

一致 Roe 代数中几何理想的刚性

Rigidity for geometric ideals in uniform Roe algebras

本文研究有界几何离散度量空间的一致 Roe 代数中几何理想的刚性问题，证明了稳定同构的几何理想对应粗等价的粗空间，并探讨了 ghostly ideals 情形与开放问题。

任意一致局部有限粗空间上一致 Roe 代数的同构刚性

Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces

研究任意一致局部有限粗空间上一致 Roe 代数的同构刚性问题，通过证明\(C^*\) - 代数同构能推出空间双射粗等价，解决了该同构刚性问题。

Embeddings of uniform Roe algebras

In this paper, we study embeddings of uniform Roe algebras. Generally speaking, given metric spaces $X$ and $Y$, we are interested in which large scale geometric properties are stable under embedding of the uniform Roe algebra of $X$ into the uniform Roe algebra of $Y$.

粗空间与一致Roe代数的量子化

A quantization of coarse spaces and uniform Roe algebras

本文提出了粗空间与一致Roe代数的量子化理论，基于N. Weaver的量子关系并依赖表示的von Neumann代数，发展了量子粗空间间映射的基础理论及其对量子一致Roe代数的推论，并引入了新的支撑扩张C*-代数类。

Two versions of the uniform Roe algebras for spaces without bounded geometry

We consider two versions of the uniform Roe algebra for uniformly discrete spaces without bounded geometry and discuss some of their properties.

Ghostly ideals in uniform Roe algebras

In this paper, we investigate the ideal structure of uniform Roe algebras for general metric spaces beyond the scope of Yu's property A. Inspired by the ideal of ghost operators coming from expander graphs and in contrast to the notion of geometric ideal, we introduce a notion of ghostly ideal in a uniform Roe algebra, whose elements are locally invisible in certain directions at infinity. We show that the geometric ideal and the ghostly ideal are respectively the smallest and the largest element in the lattice of ideals with a common invariant open subset of the unit space of the coarse group…

Ideal structures and rigidity of uniform Roe algebras

In this paper, we study the rigidity of uniform Roe algebras via the ideal structures. We showed that for given metric spaces X and Y with bounded geometry, if their uniform Roe algebras are isomorphic, then they are coarse equivalent.

通过投影对Roe代数与准局部代数的K-理论比较

$K$-Theoretic Comparison of Roe and Quasi-Local Algebras via Projections

本文针对稀疏度量空间，证明了关于Roe代数与准局部代数K-理论的三项结果，给出了二者K-理论不同的首个否定实例。

一致罗代数的刚性问题

The rigidity problem for uniform Roe algebras

本文解决了一致罗代数的刚性问题，证明同构的一致罗代数对应的度量空间是双射粗等价的。

On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces

Given a coarse space $(X,\mathcal{E})$, one can define a $\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. Špakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.

Uniform Roe coronas

A uniform Roe corona is the quotient of the uniform Roe algebra of a metric space by the ideal of compact operators. Among other results, we show that it is consistent with ZFC that isomorphism between uniform Roe coronas implies coarse equivalence between the underlying spaces, for the class of uniformly locally finite metric spaces which coarsely embed into a Hilbert space. Moreover, for uniformly locally finite metric spaces with property A, it is consistent with ZFC that isomorphism between the uniform Roe coronas is equivalent to bijective coarse equivalence between some of their cofinite…

矩阵代数嵌入到万有Roe代数中

Embeddings of matrix algebras into the universal Roe algebra

本文研究万有Roe代数的嵌入性质，证明其可嵌入$\prod_n\mathbb{M}_n$，否定Ozawa定理的推广，并得到置换算子凸包刻画及可分拟对角代数的嵌入结果。

矩阵代数嵌入一致Roe代数与拟局部代数

Embeddings of matrix algebras into uniform Roe algebras and quasi-local algebras

本文解决了矩阵代数的$\ell_\infty$-直和能否嵌入一致Roe代数与拟局部代数的问题，证明前者不可嵌入而后者可嵌入，从而说明一致Roe代数到拟局部代数的包含关系可为真包含。

Roe代数的连续场的一个例子

An example of a continuous field of Roe algebras

本文在最简单的非平凡例子$X=\mathbb R$中，通过在$[0,1]$上构造$C^*$-代数的连续场，探索了空间Roe代数与其离散化一致Roe代数之间的关系。

Cartan subalgebras in uniform Roe algebras

In this paper we study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $B\subseteq A$ of $\mathrm{C}^*$-algebras is isomorphic to the canonical inclusion of $\ell^\infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for `Roe Cartans' inside uniform Roe algebras up to automorphism when $X$ coarsely embeds into Hilbert space, and up to inner automorphism when $X$ has property A.

Maximal and reduced Roe algebras of coarsely embeddable spaces

Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C*-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds in a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K-theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two re…

均匀罗氏代数的双射刚性和比较映射的单射性

Bijective rigidity of uniform Roe algebras and injectivity of the comparison map

本文研究了均匀罗氏代数同构的均匀度量空间之间的双射粗等价与HK猜想中0阶比较映射单射性之间的等价关系，并证明了0阶比较映射无条件单射性及在空间粗连通时的分裂单射性。

A generalization of expander graphs and local reflexivity of uniform Roe algebras

We introduce a generalization of expander graphs, which is called a weak expander sequence. It is proved that a uniform Roe algebra of a weak expander sequence is not locally reflexive. It follows that uniform Roe algebras of expander graphs are not exact. We introduce the notion of a generalized box space to discuss box spaces and expander sequences in a unified framework. Key tools for the proof are amenable traces and measured groupoids associated to generalized box spaces.

K-theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations

Let $X$ be a discrete metric space with bounded geometry. We show that if $X$ admits an "A-by-CE coarse fibration", then the canonical quotient map $λ: C^*_{\max}(X)\to C^*(X)$ from the maximal Roe algebra to the Roe algebra of $X$, and the canonical quotient map $λ: C^*_{u, \max}(X)\to C^*_u(X)$ from the maximal uniform Roe algebra to the uniform Roe algebra of $X$, induce isomorphisms on $K$-theory. A typical example of such a space arises from a sequence of group extensions $\{1\to N_n\to G_n\to Q_n\to 1\}$ such that the sequence $\{N_n\}$ has Yu's property A, and the sequence $\{Q_n\}$ adm…

次指数增长一致 Roe 代数的谱不变稠密子代数

On Spectral Invariant Dense Subalgebras of Uniform Roe Algebras with Subexponential Growth

针对满足性质 P 的次指数增长群，构造 R^∞(G) 子代数，并利用容许权证明其在一致 Roe 代数 C_u^*(G) 中的谱不变性，从而把相关结果推广到多项式增长之外。

On Kuiper type theorems for uniform Roe algebras

Generalizing the case of an infinite discrete metric space of finite diameter, we say that a discrete metric space $(X,d)$ is a Kuiper space, if the group of invertible elements of its uniform Roe algebra is norm-contractible. Various sufficient conditions on $(X,d)$ to be or not to be a Kuiper space are obtained.

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

We show that if $X$ and $Y$ are uniformly locally finite metric spaces whose uniform Roe algebras, $\cstu(X)$ and $\cstu(Y)$, are isomorphic as \cstar-algebras, then $X$ and $Y$ are coarsely equivalent metric spaces. Moreover, we show that coarse equivalence between $X$ and $Y$ is equivalent to Morita equivalence between $\cstu(X)$ and $\cstu(Y)$. As an application, we obtain that if $Γ$ and $Λ$ are finitely generated groups, then the crossed products $\ell_\infty(Γ)\rtimes_rΓ$ and $ \ell_\infty(Λ)\rtimes_rΛ$ are isomorphic if and only if $Γ$ and $Λ$ are bi-Lipschitz equivalent.

Roe双模的塔尔斯基连续场

A tautological continuous field of Roe bimodules

本文将C*-代数的连续场推广为Hilbert C*-双模，通过构造特定的紧致化空间γP，得到一个塔尔斯基连续场，研究了粗等价类与Roe双模的关系。

Finite-dimensional approximation properties for uniform Roe algebras

We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C$^*_u(X)$. We prove that exactness and local reflexivity of C$^*_u(X)$ also characterize property A of $X$.

Coarse quotients of metric spaces and embeddings of uniform Roe algebras

We study embeddings of uniform Roe algebras which have "large range" in their codomain and the relation of those with coarse quotients between metric spaces. Among other results, we show that if $Y$ has property A and there is an embedding $Φ:\mathrm{C}^*_u(X)\to \mathrm{C}^*_u(Y)$ with "large range" and so that $Φ(\ell_\infty(X))$ is a Cartan subalgebra of $\mathrm{C}^*_u(Y)$, then there is a bijective coarse quotient $X\to Y$. This shows that the large scale geometry of $Y$ is, in some sense, controlled by the one of $X$. For instance, if $X$ has finite asymptotic dimension, so does $Y$.

有限秩自由阿贝尔子群生成的粗结构的一致 Roe 代数的 K-理论

The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups

本文利用 Proietti--Yamashita 谱序列，计算了由有限秩自由阿贝尔子群生成的粗结构的一致 Roe 代数的 K-群，并将其表示为群同调。

作为度量空间的拟可数逆半群，以及局部有限逆半群的一致 Roe 代数

Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

本文为拟可数逆半群构造典范真右次不变度量并定义一致 Roe 代数，证明渐近维数0等价于局部有限及强拟对角性，并刻画更弱的有限 Roe 代数条件。

Roe 代数的连续场

A continuous field of Roe algebras

本文研究了度量测度空间中受控 Delone 子集的紧致空间，通过赋予其一致 Roe 代数与 Spakula 版本的 Roe 代数，构建了重言式 $C^*$-代数族，并证明了在无孤立点的有界几何空间上可形成连续场。

Low dimensional properties of uniform Roe algebras

The goal of this paper is to study when uniform Roe algebras have certain $C^*$-algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as low dimensional, and connect these to low dimensionality of the underlying space in the sense of the asymptotic dimension of Gromov. Some of these results (for example, on stable rank one, cancellation, strong quasidiagonality, and finite decomposition rank) give definitive characterizations, while others (on real rank zero) are only partial and leave a lot o…

Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras

We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated to a uniformly locally finite metric space $X$. Under weak assumptions, these $\mathrm{C}^*$-algebras contain embedded copies of $\prod_{k}\mathrm{M}_{n_k}(\mathbb C)$ for any \emph{bounded} countable (possibly finite) collection $(n_k)_k$ of natural numbers; we aim to show that they cannot contain any other von Neumann algebras. One of our main results shows that $L_\infty[0,1]$ does not embed into any of those algebras, even by a not-necessarily-normal $*$-homomorphism. In partic…

两个 $\ell^p$ Roe 型代数的 Morita 等价

Morita equivalence of two $\ell^p$ Roe-type algebras

本文证明有界几何度量空间的 $\ell^p$ 一致 Roe 代数与 $\ell^p$ 一致代数在 Lafforgue 意义下 Morita 等价，从而具有相同 $K$-理论，并构造了不总满射的 $\ell^p$ 一致粗装配映射。

A quasi-local characterisation of $L^p$-Roe algebras

Very recently, Špakula and Tikuisis provide a new characterisation of (uniform) Roe algebras via quasi-locality when the underlying metric spaces have straight finite decomposition complexity. In this paper, we improve their method to deal with the $L^p$-version of (uniform) Roe algebras for any $p\in [1,\infty)$. Due to the lack of reflexivity on $L^1$-spaces, some extra work is required for the case of $p=1$.

通过粗几何模构建粗空间的Roe代数

Roe algebras of coarse spaces via coarse geometric modules

本文通过引入粗几何模与粗支集，将度量空间的Roe代数推广到一般粗空间，统一处理一致、非一致Roe代数及拟局部算子代数，并证明Roe代数的K-理论对粗空间具有函子性。

Property A, partial translation structures and uniform embeddings in groups

We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C*-algebra C*(T) associated with it which is a subalgebra of the uniform Roe algebra C*_u(X). We introduce a coarse invariant of the metric which provides an obstruction to embedding the space in a group. When the space is sufficiently group-like, as determined by our invariant, properties of the Roe algebra can be deduced from those of C*(T). We also give a proof of the fact that the uniform Roe algebra of a metric space is a coarse invariant up to Morita equivalence.

将图同调映射到Roe代数的$K$-理论

Mapping graph homology to $K$-theory of Roe algebras

该研究针对图的同调理论与顶点度量空间的（一致）Roe代数$K$-理论，构建了二者间的自然映射及其一致版本，并证明当图为整数集的Cayley图时，该映射为同构。

Amenability and uniform Roe algebras

Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^*(X)$ is algebraically amenable (3) $C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not pro…

Operator norm localization property for equi-approximable families of projections

The rigidity problem for uniform Roe algebras was recently positively solved. Before its solution was found, there were positive solutions under the assumption of certain technical geometric conditions. In this paper, we introduce weaker versions of the operator norm localization property (ONL) which turn out to characterize those technical geometric conditions. We use this to obtain new rigidity results for nonmetrizable coarse spaces. As an application, we provide a novel partial answer to a question of White and Willett about Cartan subalgebras of uniform Roe algebras. We also study embeddi…

一致Roe代数上的KMS态

KMS states on uniform Roe algebras

本文研究一致Roe代数上一类自然流的KMS态，证明其通过对角算子分解，并揭示了强连续态的唯一性及其与Higson冠的联系，最后应用于$n$-分支树。

极大Roe代数的Lichnerowicz消没定理

A Lichnerowicz Vanishing Theorem for the Maximal Roe Algebra

本文证明了在适当条件下，Dirac算子的极大高阶指标在极大等变Roe代数的K理论中消没，其关键是在极大等变一致Roe代数中建立泛函演算。

核维数的超限扩张

Transfinite Extension of Nuclear Dimension

本文引入$C^*$-代数的超限核维数概念，证明均匀Roe代数的超限核维数受底层空间超限渐近维数约束，并由此推出渐近性质C空间的均匀Roe代数具有corona分解性质。

Rigidity of $\ell^p$ Roe-type algebras

We investigate the rigidity of the $\ell^p$ analog of Roe-type algebras. In particular, we show that if $p\in[1,\infty)\setminus\{2\}$, then an isometric isomorphism between the $\ell^p$ uniform Roe algebras of two metric spaces with bounded geometry yields a bijective coarse equivalence between the underlying metric spaces, while a stable isometric isomorphism yields a coarse equivalence. We also obtain similar results for other $\ell^p$ Roe-type algebras. In this paper, we do not assume that the metric spaces have Yu's property A or finite decomposition complexity.

A Gelfand-type duality for coarse metric spaces with property A

We prove the following two results for a given uniformly locally finite metric space with Yu's property A: 1) The group of outer automorphisms of its uniform Roe algebra is isomorphic to its group of bijective coarse equivalences modulo closeness. 2) The group of outer automorphisms of its Roe algebra is isomorphic to its group of coarse equivalences modulo closeness. The main difficulty lies in the latter. To prove that, we obtain several uniform approximability results for maps between Roe algebras and use them to obtain a theorem about the `uniqueness' of Cartan masas of Roe algebras. We fi…

The uniform Roe algebra of an inverse semigroup

Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we can equip $S$ with a natural metric, given by the path metric in the disjoint union of its Schützenberger graphs. This graph, which we denote by $Λ_S$, inherits much of the structure of $S$. In this article we compare the C*-algebra $\mathcal{R}_S$, generated by the left regular representation of $S$ on $\ell^2(S)$ and $\ell^\infty(S)$, with the uniform Roe algebra over the metric space, namely $C^*_u(Λ_S)$. This yields a chacterization of when $\mathcal{R}…

Relative commutant pictures of Roe algebras

Let X be a proper metric space, which has finite asymptotic dimension in the sense of Gromov (or more generally, straight finite decomposition complexity of Dranishnikov and Zarichnyi). New descriptions are provided of the Roe algebra of X: (i) it consists exactly of operators which essentially commute with diagonal operators coming from Higson functions (that is, functions on X whose oscillation tends to 0 at infinity) and (ii) it consists exactly of quasi-local operators, that is, ones which have finite epsilon propogation (in the sense of Roe) for every epsilon>0. These descriptions hold bo…

$\mathbb{Z}^n$ 上均匀 Roe 代数的酉群的同伦型

Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^n$

本文研究 $\mathbb{Z}^n$ 上均匀 Roe 代数酉群的同伦型，证明其稳定化映射是同伦等价，并在 $n=1,2$ 时将其同伦型分解为 Roe 代数酉群与有理 Eilenberg--MacLane 空间的乘积。

有界几何度量空间的C*-刚性

C*-rigidity of bounded geometry metric spaces

本文证明了一致局部有限度量空间若Roe代数同构则粗等价，并得出其Roe代数外自同构群典范同构于空间粗等价群的结论。

On Banach algebras of band-dominated operators and their order structure

The goal of this paper is to study band-dominated operators on Banach spaces with Schauder basis with respect to uniformly locally finite metric spaces as well as the Banach algebras generated by them: the so called uniform Roe algebras. We investigate several kinds of isomorphisms between those Banach spaces (e.g., isomorphisms preserving norm, order, algebraic structure, etc) and prove several rigidity results on when a certain kind of isomorphism between the uniform Roe algebras implies that the base metric spaces are (bijectively) coarsely equivalent.

Roe 类代数中极限算子与 Fredholm 性的一种非标准分析方法

A nonstanadard analysis approach to limit operators and Fredholmness in Roe-like algebras

本文利用非标准分析定义遥远星系上的极限算子，给出一致 Roe 或拟局部代数中广义 Fredholm 性及性质 A 下 Fredholm 性的可逆性刻画，并以更少极限算子强化既有结果。

Coarse Baum-Connes conjecture and rigidity for Roe algebras

In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if $X$ and $Y$ are two uniformly locally finite metric spaces such that their Roe algebras are $*$-isomorphic, then $X$ and $Y$ are coarsely equivalent provided either $X$ or $Y$ satisfies the coarse Baum-Connes conjecture with coefficients. It is well-known that coarse embeddability into a Hilbert space implies the coarse Baum-Connes conjecture with coefficients. On the other hand, we provide a new example of a finitely generated group satisfying the coarse Baum-C…

关于精确离散群和$\ell^p$一致Roe代数的一些$p$-逼近性质

On some $p$-approximation properties of discrete groups and $\ell^p$ uniform Roe algebras

研究$\ell^p$一致Roe代数的$p$-逼近性质及其与粗几何和群论的联系，引入$p$-不变平移逼近性质，并证明精确群与$p$-算子逼近性质的等价性。

均匀Roe代数上的流

Flows on uniform Roe algebras

本文研究均匀Roe代数上的粗流，证明其可由自伴算子表示，并在性质A下给出粗流的充要条件及余循环扰动与共轭的刻画。

Going-Down functors and the Künneth formula for crossed products by étale groupoids

We study the connection between the Baum-Connes conjecture for an ample groupoid $G$ with coefficient $A$ and the Künneth formula for the K-theory of tensor products by the crossed product $A\rtimes_r G$. To do so we develop the machinery of Going-Down functors for ample groupoids. As an application we prove that both the uniform Roe algebra of a coarse space which uniformly embeds into a Hilbert space and the maximal Roe algebra of a space admitting a fibred coarse embedding into a Hilbert space satisfy the Künneth formula. We also provide a stability result for the Künneth formula using cont…

离散可逆群的一致Roe代数中的严格比较性质

Strict comparison holds in the uniform Roe algebra of a discrete amenable group

本文证明离散可逆群的一致Roe代数中，正元素迹态下的严格不等式蕴含Cuntz次等价，并证明其通用极小集交叉积C*-代数是强AH代数。